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158 records · Page 9

All sources NASA NTRS DOE OSTI

DOE OSTI2024

DG-IMEX method for a two-moment model for radiation transport in the $\mathscr{O}$($v$/$c$) limit

Here, we consider neutral particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to $\mathscr{O}$($v$/$c$), expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. The model is conservative for the correct $\mathscr{O}$($v$/$c$) Eulerian-frame number density and is consistent, to $\mathscr{O}$($v$/$c$), with Eulerian-frame energy and momentum conservation. This model is closely related to the one promoted by Lowrie et al. and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The proposed numerical method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical uxes; (iii) a realizability-preserving implicit collision update; and(iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed global convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose a "spectral redistribution" scheme that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1994

Damage-Mitigating Control of Space Propulsion Systems for High Performance and Extended Life

A major goal in the control of complex mechanical system such as spacecraft rocket engine's advanced aircraft, and power plants is to achieve high performance with increased reliability, component durability, and maintainability. The current practice of decision and control systems synthesis focuses on improving performance and diagnostic capabilities under constraints that often do not adequately represent the materials degradation. In view of the high performance requirements of the system and availability of improved materials, the lack of appropriate knowledge about the properties of these materials will lead to either less than achievable performance due to overly conservative design, or over-straining of the structure leading to unexpected failures and drastic reduction of the service life. The key idea in this report is that a significant improvement in service life could be achieved by a small reduction in the system dynamic performance. The major task is to characterize the damage generation process, and then utilize this information in a mathematical form to synthesize a control law that would meet the system requirements and simultaneously satisfy the constraints that are imposed by the material and structural properties of the critical components. The concept of damage mitigation is introduced for control of mechanical systems to achieve high performance with a prolonged life span. A model of fatigue damage dynamics is formulated in the continuous-time setting, instead of a cycle-based representation, for direct application to control systems synthesis. An optimal control policy is then formulated via nonlinear programming under specified constraints of the damage rate and accumulated damage. The results of simulation experiments for the transient upthrust of a bipropellant rocket engine are presented to demonstrate efficacy of the damage-mitigating control concept.

Ray, Asok↗

NASA NTRS1987

Extension of a semi-implicit shock-capturing algorithm for 3-D fully coupled, chemically reacting flows in generalized coordinates

A semiimplicit high-resolution shock-capturing method for multidimensional systems of hyperbolic conservation laws with stiff source terms has been developed by Yee and Shinn (1987). The goal of this work is to extend this method to solve the three-dimensional fully coupled Navier-Stokes equations for a hypersonic chemically reacting flow in generalized coordinates. In this formulation, the global continuity equation was replaced by all the species continuity equations. The shock-capturing technique is a second-order-accurate, symmetric total-variation-diminishing method which accounts fully and directly for the coupling among the fluid and all the species. To verify the current approach, it was implemented into an existing computer code which contained the MacCormack method. Test results for a five-species reacting flow are shown to be oscillation-free around the shock, and the time spent per iteration only doubles when compared to the result using classical way of supplying numerical dissipation. The extra computation is more than justified by the elimination of spurious oscillation and nonlinear instability associated with the classical shock-capturing schemes in computing hypersonic reacting flows.

Shinn, J. L.↗

NASA NTRS1998

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized demoralization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin demoralization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS, and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobean linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Discrete maximum principle theory will be presented for general finite volume approximations on unstructured meshes. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc, will. be addressed as needed.

Barth, Timothy J.↗

DOE OSTI2021

Explicit structure-preserving geometric particle-in-cell algorithm in curvilinear orthogonal coordinate systems and its applications to whole-device 6D kinetic simulations of tokamak physics

Explicit structure-preserving geometric particle-in-cell (PIC) algorithm in curvilinear orthogonal coordinate systems is developed. The work reported represents a further development of the structure-preserving geometric PIC algorithm achieving the goal of practical applications in magnetic fusion research. The algorithm is constructed by discretizing the field theory for the system of charged particles and electromagnetic field using Whitney forms, discrete exterior calculus, and explicit non-canonical symplectic integration. In addition to the truncated infinitely dimensional symplectic structure, the algorithm preserves exactly many important physical symmetries and conservation laws, such as local energy conservation, gauge symmetry and the corresponding local charge conservation. As a result, the algorithm possesses the long-term accuracy and fidelity required for first-principles-based simulations of the multiscale tokamak physics. The algorithm has been implemented in the SymPIC code, which is designed for high-efficiency massively-parallel PIC simulations in modern clusters. The code has been applied to carry out whole-device 6D kinetic simulation studies of tokamak physics. A self-consistent kinetic steady state for fusion plasma in the tokamak geometry is numerically found with a predominately diagonal and anisotropic pressure tensor. The state also admits a steady-state sub-sonic ion flow in the range of 10 km s -1 , agreeing with experimental observations and analytical calculations Kinetic ballooning instability in the self-consistent kinetic steady state is simulated. It is shown that high-n ballooning modes have larger growth rates than low-n global modes, and in the nonlinear phase the modes saturate approximately in 5 ion transit times at the 2% level by the E × B flow generated by the instability. These results are consistent with early and recent electromagnetic gyrokinetic simulations.

43 PARTICLE ACCELERATORS↗

NASA NTRS1997

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized discretization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin discretization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobian linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. These variants have been implemented in the "ELF" library for which example calculations will be shown. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Some prevalent limiting strategies will be reviewed. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc. will be addressed as needed.

Barth, Timothy↗

NASA NTRSDate not supplied

A Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Reacting Hypersonic Flows

The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced, is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this talk. Moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) [1],[2] are then presented for solving compressible flow problems with discontinuities based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. In the MDG-ICE formulation, both conservative quantities and grid geometry are considered as independent variables. A space-time DG formulation is used to solve the multi-material compressible Euler equations in the standard discontinuous solution space and the discrete grid geometry is solved using a variational formulation in a continuous space. A self-adaptive Levenberg-Marquardt method is utilized to solve the resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation. A number of numerical experiments for a variety of flow problems are conducted to assess the accuracy and performance of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to deliver the designed order of both h- and p-convergence even for discontinuous solutions, and detect all types of interfaces, via interface condition enforcement and satisfy, via grid movement, the compressible Euler equations and the associated interface condition.

Discontinuous Galekin↗

DOE OSTI2024

Conservation law for electromagnetic fields in a space-time-varying medium and its implications

We discuss the propagation of electromagnetic waves in a space-time modulated medium, for which neither energy nor momentum remains conserved. We have found a new conservation law when modulation moves at a constant speed v in a traveling-wave fashion. We use this law to study how light is scattered inside such a medium. We consider both the subluminal (v < c) andsuperluminal (v > c) cases and find that they differ considerably in terms of what remains conserved. We show that the total number of photons is conserved in the subluminal case. In contrast, it is the difference between the forward and backward moving photons that remains conserved in the superluminal case. We also study the reciprocity issue for a space-time modulated medium and found that reciprocity holds in all situations. Here, we develop a quantum formulation of our scattering problem and show that the subluminal case is similar to the action of a beam splitter. In the superluminal case, photon pairs can be generated from vacuum in a fashion analogous to the two-mode frequency down-conversion process.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1993

A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations

A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel width, depending on the Reynolds number.

Scott, James R.↗

DOE OSTI2023

Recipes for when physics fails: recovering robust learning of physics informed neural networks

Abstract Physics-informed neural networks (PINNs) have been shown to be effective in solving partial differential equations by capturing the physics induced constraints as a part of the training loss function. This paper shows that a PINN can be sensitive to errors in training data and overfit itself in dynamically propagating these errors over the domain of the solution of the PDE. It also shows how physical regularizations based on continuity criteria and conservation laws fail to address this issue and rather introduce problems of their own causing the deep network to converge to a physics-obeying local minimum instead of the global minimum. We introduce Gaussian process (GP) based smoothing that recovers the performance of a PINN and promises a robust architecture against noise/errors in measurements. Additionally, we illustrate an inexpensive method of quantifying the evolution of uncertainty based on the variance estimation of GPs on boundary data. Robust PINN performance is also shown to be achievable by choice of sparse sets of inducing points based on sparsely induced GPs. We demonstrate the performance of our proposed methods and compare the results from existing benchmark models in literature for time-dependent Schrödinger and Burgers’ equations.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2000

The Finer Details: Climate Modeling

If you want to know whether you will need sunscreen or an umbrella for tomorrow's picnic, you can simply read the local weather report. However, if you are calculating the impact of gas combustion on global temperatures, or anticipating next year's rainfall levels to set water conservation policy, you must conduct a more comprehensive investigation. Such complex matters require long-range modeling techniques that predict broad trends in climate development rather than day-to-day details. Climate models are built from equations that calculate the progression of weather-related conditions over time. Based on the laws of physics, climate model equations have been developed to predict a number of environmental factors, for example: 1. Amount of solar radiation that hits the Earth. 2. Varying proportions of gases that make up the air. 3. Temperature at the Earth's surface. 4. Circulation of ocean and wind currents. 5. Development of cloud cover. Numerical modeling of the climate can improve our understanding of both the past and, the future. A model can confirm the accuracy of environmental measurements taken. in, the past and can even fill in gaps in those records. In addition, by quantifying the relationship between different aspects of climate, scientists can estimate how a future change in one aspect may alter the rest of the world. For example, could an increase in the temperature of the Pacific Ocean somehow set off a drought on the other side of the world? A computer simulation could lead to an answer for this and other questions. Quantifying the chaotic, nonlinear activities that shape our climate is no easy matter. You cannot run these simulations on your desktop computer and expect results by the time you have finished checking your morning e-mail. Efficient and accurate climate modeling requires powerful computers that can process billions of mathematical calculations in a single second. The NCCS exists to provide this degree of vast computing capability.

Source record↗

DOE OSTI2020

PPINN: Parareal physics-informed neural network for time-dependent PDEs

Physics-informed neural networks (PINNs) encode physical conservation laws and prior physical knowledge into the neural networks, ensuring the correct physics is represented accurately while alleviating the need for supervised learning to a great degree. While effective for relatively short-term time integration, when long time integration of the time-dependent PDEs is sought, the time–space domain may become arbitrarily large and hence training of the neural network may become prohibitively expensive. To this end, we develop a parareal physics-informed neural network (PPINN), hence decomposing a long-time problem into many independent short-time problems supervised by an inexpensive/fast coarse-grained (CG) solver. In particular, the serial CG solver is designed to provide approximate predictions of the solution at discrete times, while initiate many fine PINNs simultaneously to correct the solution iteratively. There is a two-fold benefit from training PINNs with small-data sets rather than working on a large-data set directly, i.e., training of individual PINNs with small-data is much faster, while training the fine PINNs can be readily parallelized. Consequently, compared to the original PINN approach, the proposed PPINN approach may achieve a significant speed-up for long-time integration of PDEs, assuming that the CG solver is fast and can provide reasonable predictions of the solution, hence aiding the PPINN solution to converge in just a few iterations. To investigate the PPINN performance on solving time-dependent PDEs, we first apply the PPINN to solve the Burgers equation, and subsequently we apply the PPINN to solve a two-dimensional nonlinear diffusion–reaction equation. Furthermore, our results demonstrate that PPINNs converge in a few iterations with significant speed-ups proportional to the number of time-subdomains employed.

42 ENGINEERING↗

DOE OSTI2025

Lectures on statistical mechanics

Presented here is a transcription of the lecture notes from Professor Allan N. Kaufman’s graduate statistical mechanics course Physics 212A and 212B at the University of California Berkeley from the 1972–1973 academic year. 212A addressed equilibrium statistical mechanics with topics: fundamentals (micro-canonical and sub-canonical ensembles, adiabatic law and action conservation, fluctuations, pressure, and virial theorem), classical fluids and other systems (equation of state, deviations from ideality, virial coefficients and van der Waals potential, canonical ensemble and partition function, quasistatic evolution, grand-canonical ensemble and partition function, chemical potential, simple model of a phase transition, quantum virial expansion, numerical simulation of equations of state, and phase transition), chemical equilibrium (systems with multiple species and chemical reactions, law of mass action, Saha equation, chemical equilibrium including ionization and excited states), and long-range interactions (including Coulomb, dipole, and gravitational interactions, Debye–Hückel theory, and shielding). 212B addressed nonequilibrium statistical mechanics with topics: fundamentals (definitions: realizations, moments, characteristic function, and discrete variables), Brownian motion (Langevin equation, fluctuation–dissipation theorem, spatial diffusion, Boltzmann’s H-theorem), Liouville and Klimontovich equations, Landau equation (derivation, elaboration, and H-theorem, and irreversibility), Markov processes and Fokker–Planck equation (derivations of the Fokker–Planck equation and a master equation), linear response and transport theory (linear Boltzmann equation, linear response theory of Kubo and Mori, relation of entropy production to electrical conductivity, transport relations and coefficients, normal mode solutions of the transport equations, sketch of a generalized Langevin equation method for transport theory), and an introduction to nonequilibrium quantum statistical mechanics.

plasma dynamics↗

DOE OSTI2022

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows

Herein we demonstrate that the reformulation of renormalization group (RG) flow equations as nonlinear heat equations has severe implications on the understanding of RG flows in general. We demonstrate by explicitly constructing an entropy function for a zero-dimensional Z 2 -symmetric model that the dissipative character of generic nonlinear diffusion equations is also hard-coded in the functional RG equation. This renders RG flows manifestly irreversible, revealing the semigroup property of RG transformations on the level of the flow equation itself. Additionally, we argue that the dissipative character of RG flows, its irreversibility and the entropy production during the RG flow may be linked to the existence of a so-called C– / A-function. In total, this introduces an asymmetry in the so-called RG time—in complete analogy to the thermodynamic arrow of time—and allows for an interpretation of infrared actions as equilibrium solutions of dissipative RG flows equations. The impossibility of resolving microphysics from macrophysics is evident in this framework. Furthermore, we directly link the irreversibility and the entropy production in RG flows to an explicit numerical entropy production, which is manifest in diffusive and non-linear partial differential equations (PDEs) and a standard mathematical tool for the analysis of PDEs. Using exactly solvable zero-dimensional Z 2 -symmetric models, we explicitly compute the (numerical) entropy production related to the total variation nonincreasing property of the PDE during RG flows toward the infrared limit. Finally, we discuss generalizations of our findings and relations to the C– / A-theorem as well as how our work may help to construct truncations of RG flow equations in the future, including numerically stable schemes for solving the corresponding PDEs.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗